{
  "id": "1610.01871",
  "version": "v1",
  "published": "2016-10-02T14:13:44.000Z",
  "updated": "2016-10-02T14:13:44.000Z",
  "title": "Solutions to the inexact resolvent inclusion problem with applications to nonlinear analysis and optimization",
  "authors": [
    "Daniel Reem",
    "Simeon Reich"
  ],
  "categories": [
    "math.OC",
    "math.FA"
  ],
  "abstract": "The exact resolvent inclusion problem has various applications in nonlinear analysis and optimization theory, such as devising (proximal) algorithmic schemes aiming at minimizing convex functions and finding zeros of nonlinear operators. The inexact version of this problem allows error terms to appear and hence enables one to better deal with noise and computational errors, as well as superiorization. The issue of existence and uniqueness of solutions to this problem has neither been discussed in a comprehensive way nor in a general setting. We show that if the space is a real reflexive Banach space, the inducing function is fully Legendre, and the operator is maximally monotone, then the problem admits a unique and explicit solution. We use this result to significantly extend the scope of numerous known inexact algorithmic schemes (and corresponding convergence results) in various finite and infinite dimensional settings. In the corresponding papers the question whether there exist sequences satisfying the schemes in the inexact case (in which many of these schemes have a strongly implicit nature) has not been answered. As a byproduct we show, under simple conditions, the (H\\\"older) continuity of the protoresolvent and the continuous dependence (stability) of the solution of the inexact resolvent inclusion problem on the initial data.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-02T14:13:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "90C31",
      "47H05",
      "47J25",
      "90C30",
      "49M37",
      "F.2.1",
      "G.1.0"
    ],
    "keywords": [
      "inexact resolvent inclusion problem",
      "nonlinear analysis",
      "optimization",
      "applications",
      "infinite dimensional settings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}